π in Other Universes

&gt; Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable<p>This is a really, <i>really</i> nice expression of something my mind&#x27;s been hovering around for a while.

Note that even if another universe has a different π when it comes to geometry they are still going to also have an important constant that has the same value as our π.<p>E.g., the zeros of the function defined by the series x - x^3&#x2F;3! + x^5&#x2F;5! - x^7&#x2F;7! + ... are nπ where n is an integer and π is our π. Another place our pi will come up is in the exponential function. It&#x27;s periodic with period 2πi.

One thing this doesn&#x27;t touch on is that there are multiple meaningful definitions of pi-like constants for the p-norm unit circle that don&#x27;t necessarily agree with each other in p != 2. Defining pi as the area of the unit circle gives an entirely different set of values that satisfying some wonderful properties - in particular, that definition of pi turns out to be the periodicity constant for a (arguably) natural set of trigonometric functions for the p-circle. Furthermore, pi(p) = 2 Beta(1&#x2F;p,1&#x2F;p)&#x2F;p...<p>However, this (circumference&#x2F;arc-length based) definition of pi does have a fascinating property for conjugate p,q: pi(p) = pi(q)<p>&quot;Squigonometry: The Study of Imperfect Circles&quot; is a very fun reference for this sort of stuff.

* pi = 3.14159… appears in analysis and by extension statistics, independent of geometry. So aliens in these other universes would know this value, they’d just have a different constant for circles. Since they wouldn’t use Greek letters anyway, we’d have to translate, and it would be a bit silly to equate their 3.757… with “pi” instead of their 3.14159…<p>* Personal aside: Of course, whether 3.14… (pi), 6.28… (2pi) or even 0.785… (pi&#x2F;4) should be the fundamental constant is debatable, and aliens might have different ideas about that.<p>* The article introduces the concept of metrics to explain that there could be different circle constants in other universes. But arbitrary metrics don’t necessarily have linear scaling or translation invariance. You need stronger assumptions than a metric to meaningfully define a circle <i>constant</i> at all, like a normed vector space. AFAICT, all of the given examples are in fact normed vector spaces, not just metric spaces.

This person is not a sailor. Sailing orthogonal to the wind, a &quot;beam reach&quot;, is the fastest point of sail due to the lift of the sail.

All of these assume your background metric is Euclidean.<p>If your background 2D metric is a projection of a warped 3D space, you can make π as big as you want by tugging on the centre of the circle.

When I was a kid I liked to muse about relationships like these. Since I was a kid I imagined that there might have been a god that created the universe, and imagined that they were a bored kid like me perhaps making it as a school assignment.<p>So what if the god had turned the pi or e knobs to a rational number (presumably in a god’s universe knobs can be turned to precise irrational values). Would it have made our lives easier or harder (probably easier…?). Or what about the apparent size of earth&#x2F;moon&#x2F;sun when viewed from earth? It’s a great clue, but perhaps we would have known more about astronomy if that coincidence had not existed? (We would have missed out on that fabulous Connie Willis story though).<p>Maybe all those weird cosmological QM oddities and (literally obscure) imbalances needing mysterious dark matter are just due to bugs in a kid’s rushed assignment and actually don’t make sense?<p>But the irrationals…they led to the most musing.

<i>IF</i> I&#x27;ve correctly assesed the zeitgeist of HN postings<p><i>THEN</i> it follows Terence Tao&#x27;s Introduction to Measure Theory must be a bullet.<p><a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=38064211">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=38064211</a><p>But seriously, who&#x27;s going to read|skim a free 260+ tract on measure theory?<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Measure_(mathematics)" rel="nofollow noreferrer">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Measure_(mathematics)</a>

There&#x27;s this fun space made of p-adic numbers upon which you can define a simple distance, and then circles have mind bending properties like the diameter (max edge to edge distance) and radius (distance from edge to center) being equal to each other.<p>Quirky stuff happens to disc area and perimeter as well, and open discs are also closed. The equivalent of Pi there is nuts.<p>Sadly I can&#x27;t recall the details (it was a 2000-ish exercise on my maths course).<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;P-adic_number#Topological_properties" rel="nofollow noreferrer">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;P-adic_number#Topological_prop...</a>

The boat analogy seems particularly poor.<p>a) Comparing a sailboat on a windy day to a sail boat on an [implied] non-windy day? Surely the boat with no wind wouldn&#x27;t even have a circle.<p>b) I&#x27;m no boatologist, but if the wind is X knots, then the boat can travel downwind at a rate of X knots, but contrary to what the article states, the boat would be able to travel cross-winds at some multiple of X. So you would get something resembling an oval, but in the opposite orientation as depicted.<p>Also, it&#x27;s worth pointing out that it&#x27;s perfectly possible for a boat to travel &quot;into&quot; the wind via &quot;tacking and jibing&quot;

OMG. After all this time, you&#x27;re telling me the drafters of the Indiana Pi Bill [0] could have been right all along?<p>It would mean that Indiana happened to be in a different Universe at the time, were:<p><pre><code> d=1&#x2F;(2 √3) ∑( n=1…6 )∣∣x sin(3πn )+y cos(3πn )∣∣ [1] </code></pre> Well, whose to say otherwise?<p>[0] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Indiana_Pi_Bill" rel="nofollow noreferrer">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Indiana_Pi_Bill</a> [1] Poor man representation of the same equation in the article.

Maybe worth pointing out that there are countless other weird Universes where &quot;Pi&quot; retains its standard value.<p>This is the domain of differential geometry where the relation of circumference and radius holds only in the limit of infinitesimally small.<p>By all accounts our own Universe is of such a deformed-in-the-large but Euclidean-in-the-small variety. At least for as far we understand geometry in the quantum realm.

This largely depends on how one defines pi. I believe that the concept of R^n (Euclidean space) exists even in entirely different physical spaces. This is because Euclidean space represents a universally recognized idea of simple space in terms of curvature. For instance, in any world, the concept of &#x27;0&#x27; represents simplicity. In this context, pi will always remain constant.

I noticed that all the &quot;circles&quot; for alternative metrics are aligned with the coordinate system. For example, the one for the Manhattan distance has its corners on the coordinate axes.<p>What if we added an additional condition that a distance metric should not change when the orientation of the coordinate system is changed? Could we still have different values for the pi constant then?

Not sure about your universe, but here on earth, pi is 2. The length of the equator is 4 times the distance from the pole. (Approx.)

I must be missing something. In the example of using a sailboat with constant wind and distance, wouldn’t sailing against the wind (let’s call it any constant oppositional force), cause us to get a circle, just shifted from the origin? Not an ellipsis?

A p-norm of 3 make quit a funky &quot;retro&quot; rounded corner style for avatars. It also shows what the &quot;opposite&quot; of rounded corners might look like. In CSS you can only go to a circle.

This is the sort of thing that makes me want to learn VR development.

A circle is a pillar for a 3 dimensional universe in a 2 dimensional universe. So I guess every dimensional jump has one and the binary one is the origin constant?

The hexagonal metric at the end uses pi in its definition - is this our value of pi, or the value of 3 that that metric provides?

The area of the circle in Manhattan distance comes out to 2 million, but pi * r^2 is 4 million. What am I doing wrong?

Excellent article, both informative and accessible, and the interactive visualizations are lovely.

Didn&#x27;t 3blue1brown have a video on exactly this?